Then after extracting something the software iTunes says it is verifying restore with Apple... then I get the error "the iPhone could not be restored. This device isnt eligible for the requested build."

Weird.
Its trying to communicate with Apples servers to verify the restore but something is blocking it.
You got a firewall or any kind of IP blocking, security suite program running?
Try restoring it on another computer if you can if this one is still giving you problems.

Then after extracting something the software iTunes says it is verifying restore with Apple... then I get the error "the iPhone could not be restored. This device isnt eligible for the requested build."

Download the Firmware (IPSW) from here for 4.3. Then open iTunes and Shift+Click restore and select the IPSW you downloaded. See if that works.

If that doesn't work download Tinyumbrella from here. Open TU and click Advanced and make sure "Set Hosts to Cydia on Exit" is unchecked and hit apply and close TU. Try doing the Restore again and see if that works.

ETA: If neither of those work open TU again and click save SHSH with your iPhone connected after if finishes and it shows 4.3 in the list Click "Start TSS server" and try doing the restore again. If it starts the restore it may just be that the apple servers are down or something.

http://www.iclarified.com/entry/index.php?enid=1034
Very weird.
You tried another computer and its giving you the same you said?
Yours must be the only iphone not beeing able to be restored or its user error.
Try what "himni" said also and maybe a DFU restore I put up above.

Have you tried to update then restore? I had similar issues and upgraded first. I then downloaded the latest iTunes and restored. I'm thinking downloading the latest iTunes would have let me restore first but I'll never know. If you're already running 10.2 then try the upgrade then restore.

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